Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through
$$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$
and ...

The theory of a real (hermitian) scalar field can be found in many books and everywhere online. On the other hand, if we take the field non-hermitian, then I can only find notes on path integrals. I ...

I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved.
While I'm at it, I might as well add that I'd be most interested ...

I'm trying to figure out how one would write down the hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...

Would $[\hat{Q},\hat{H}]$ correspond to an observable?
Where $\hat{Q}$ is an observable and $\hat{H}$ is the Hamiltonian.
Surely that would just mean that $[\hat{Q},\hat{H}]$ would commute i.e. = 0?:
...

My understanding of the Jacobi energy function $h$ as defined in Goldstein is that it is the total energy $T+V$ expressed as,
\begin{equation}
h(q,\dot q,t)=\sum \frac{\partial L}{\partial \dot q}\dot ...

How do we find the phase space density from the Hamiltonian?
For example: Consider a classical gas made of N identical non-interacting particles in 1d. Each molecule is characterised by centre mass ...

I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian
$$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$
I want to express the ...

Assume we have a system in a state $\psi$ that is a superposition of eigenvectors of some observable $A$. Now we make a measurement of the observable $A$; the state after the measurement $\phi$ is a ...

In general, in quantum mechanics we can prove probability current or the Schrodinger equation and other quantities are gauge invariant. However, the Hamiltonian isn't gauge invariant. Under a gauge ...

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...

If I'm asked to prove that a system is/ isn't conservative and compare it to whether or not the Hamiltonian is conserved, does that mean I need to compute the time derivative of energy $(T+U)$? Doing ...

Is there a relationship between the Hamiltonian and the number of observables needed in a CSCO to fully describe a quantum system? I am thinking that a new observable is needed each time there is a ...

To find the allowed energies for a system, I can find the spectrum of the Hamiltonian $\hat{H}_{\psi}$ given a wavefunction $\psi$ representing the state of the system. 3 cases might happen: either ...

We have the Hamiltonian, a concept that was based on trajectories being used extensively in General Relativity, Electromagnetism, Quantum Mechanics, Classical Physics and lot more. Where we use the ...